$\displaystyle \left| x \right|$ , Hoplessly Positive.

Note: $\displaystyle \left| x \right|$ isn't the variable. $\displaystyle x$ is the variable.

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$\displaystyle { \left| x \right| }^{ 2 }+\left| x \right| -6=0\\ \Rightarrow \left( \left| x \right| +3 \right) \left( \left| x \right| -2 \right) =0$ .

But, $\displaystyle \left| x \right| +3\neq 0$ because $\displaystyle \left| x \right| \neq -3$ (i.e $\left| x \right|$ cannot be negative.)

Therefore, $\displaystyle \left| x \right| -2=0\\ \Rightarrow x=2\quad or\quad -2.$ .

Clearly sum of roots is $\displaystyle 2+\left( -2 \right) =0$ and product of roots is $\displaystyle 2\times \left( -2 \right) =-4$ . None of which are given in the options.

Hence the answer is (D) None of these ''.